Let \(T\) be an isosceles right triangle and let \(S_1, S_2, S_3, \dots\) be the homothetic copies of a square \(S\). In this paper, we consider the parallel covering and packing of \(T\) with the sequence \(\{S_n\}\) of squares.

A graph \(G\) is called super edge-magic if there exists a bijective function \(f: V(G) \cup E(G) \rightarrow \{1, 2, \dots, |V(G)| + |E(G)|\}\) such that \(f(V(G)) = \{1, 2, \dots, |V(G)|\}\) and \(f(u) + f(v) + f(uv)\) is a constant for each \(uv \in E(G)\). The super edge-magic deficiency, \(\mu_s(G)\), of a graph \(G\) is defined as the smallest nonnegative integer \(n\) with the property that the graph \(G \cup nK_1\) is super edge-magic, or \(+\infty\) if there exists no such integer \(n\). In this paper, the super edge-magic deficiency of certain 2-regular graphs with two components is computed, which leads us to a conjecture on the super edge-magic deficiency of graphs in this class.

From a computer search, new minimum sizes for the maximal partial spreads in \(PG(3,q)\) have been obtained for \(q = 8, 9, 16\) and for every \(q\) such that \(25 \leq q \leq 101\). Furthermore, density results in the cases \(q = 8, 9, 16, 19, 23, 25, 27\) have been obtained. Finally, the already known exceptional size \(45\) for \(q = 7\) has been found again.

We decompose the complete multigraph \(K(v, \lambda)\) into copies of a graph \(H_i\) (\(i = 1, 2, 3\)). Each \(H_i\) is a near-triangle in that it is connected and has \(3\) vertices. In several cases, the decompositions are completed using classical combinatorial sequences due to Langford and Skolem.

It may be desired to seat \(n\) people along a row (as at a lunch counter), or \(n+1\) people around a circular table, in \(n\) consecutive rounds of seating, so that each person \(x\) has every other person \(y\) on their right exactly once, and on their left exactly once, in one of the seatings. Alternatively, it may be desired to seat \(2n\) people along a row, or \(2n + 1\) people around a circular table, in only \(n\) consecutive rounds, so that each person \(x\) is adjacent to every other person \(y\) (either on the right or the left) exactly once. We show that these problems are solved using the rows of Tuscan squares to specify the successive rounds of seatings.

Let \(\mathbb{F}_q^(n+1)\) denote the \((n+l)\)-dimensional projective space over a finite field \(\mathbb{F}_q\). For a fixed integer \(m \leq \min\{n,l\}\), denote by \(\mathcal{L}_o^m(\mathbb{F}_q^{n+1})\) the set of all subspaces of type \((t,t_1)\), where \(t_1 \leq t \leq m\). Partially ordered by ordinary inclusion, one family of quasi-regular semilattices is obtained. Moreover, we compute all its parameters.

If \(X\) is a geodesic metric space and \(x_1, x_2, x_3 \in X\), a geodesic triangle \(T = \{x_1, x_2, x_3\}\) is the union of the three geodesics \([x_1x_2]\), \([x_2x_3]\) and \([x_3x_1]\) in \(X\). The space \(X\) is \(\delta\)-hyperbolic (in the Gromov sense) if any side of \(T\) is contained in a \(5\delta\)-neighborhood of the union of the two other sides, for every geodesic triangle \(T\) in \(X\). We denote by \(\delta(X)\) the sharp hyperbolicity constant of \(X\), i.e., \(\delta(X) := \inf\{\delta \geq 0: X \text{ is } \delta\text{-hyperbolic}\}\). The main result of this paper is the inequality \(\delta(G) \leq \delta(\mathcal{L}(G))\) for the line graph \(\mathcal{L}(G)\) of every graph \(G\). We prove also the upper bound \(\delta(L(G)) \leq 5\delta(G) + 3l_{\max}\), where \(\max\) is the supremum of the lengths of the edges of \(G\). Furthermore, if every edge of \(G\) has length \(k\), we obtain \(\delta(G) \leq \delta(\mathcal{L}(G)) \leq 5\delta(G) + 5k/2\).

For graphs \(G\) and \(H\), the size-balanced Ramsey multipartite number \(m_j(G, H)\) is defined as the smallest positive integer \(s\) such that any arbitrary red/blue coloring of the graph \(K_{s,s}\) forces the appearance of a red \(G\) or a blue \(H\). In the main case of this paper, we generalize methods used in finding bipartite Ramsey numbers for \(b(nK_2, mK_2)\) to finding the balanced Ramsey multipartite number \(m_j(nK_2, mK_2)\).

The subdivision graph \(S(G)\) of a graph \(G\) is the graph obtained by inserting a new vertex into every edge of \(G\). Let \(G_1\) and \(G_2\) be two vertex-disjoint graphs. The subdivision-vertex corona of \(G_1\) and \(G_2\), denoted by \(G_1 \odot G_2\), is the graph obtained from \(S(G_1)\) and \(|V(G_1)|\) copies of \(G_2\), all vertex-disjoint, by joining the \(i\)th vertex of \(V(G_1)\) to every vertex in the \(i\)th copy of \(G_2\). The subdivision-edge corona of \(G_1\) and \(G_2\), denoted by \(G_1 \ominus G_2\), is the graph obtained from \(S(G_1)\) and \(|I(G_1)|\) copies of \(G_2\), all vertex-disjoint, by joining the \(i\)th vertex of \(I(G_1)\) to every vertex in the \(i\)th copy of \(G_2\), where \(I(G_1)\) is the set of inserted vertices of \(S(G_1)\). In this paper, we determine the generalized characteristic polynomial of \(G_1 \odot G_2\) (respectively, \(G_1 \ominus G_2\)). As applications, the results on the spectra of \( G_1 \odot G_2\) (respectively, \(G_1 \ominus G_2\)) enable us to construct infinitely many pairs of \(\Phi\)-cospectral graphs. The adjacency spectra of \(G_1 \odot G_2\) (respectively, \(G_1 \ominus G_2\)) help us to construct many infinite families of integral graphs. By using the Laplacian spectra, we also obtain the number of spanning trees and Kirchhoff index of \(G_1 \odot G_2\) and \(G_1 \ominus G_2\), respectively.

In this paper, we study arc-transitive pentavalent graphs of order \(4p^n\), where \(p\) is a prime and \(n\) is a positive integer. It is proved that no such graph exists for each prime \(p \geq 5\), and all such graphs with \(p = 2\) or \(3\) which are \(G\)-basic (that is, \(G\) has no non-trivial normal subgroup such that the graph is a normal cover of the corresponding normal quotient graph) are determined. Moreover, as an application, arc-transitive pentavalent graphs of order \(4p^2\) and \(4p^3\) are determined.